Linear Algebra

Solving Linear Systems......Page 11
Gauss' Method......Page 12
Describing the Solution Set......Page 21
General = Particular + Homogeneous......Page 30
Vectors in Space......Page 42
Length and Angle Measures*......Page 48
Gauss-Jordan Reduction......Page 55
Row Equivalence......Page 61
Topic: Computer Algebra Systems......Page 71
Topic: Input-Output Analysis......Page 73
Topic: Accuracy of Computations......Page 77
Topic: Analyzing Networks......Page 82
Vector Spaces......Page 89
Definition and Examples......Page 90
Subspaces and Spanning Sets......Page 101
Definition and Examples......Page 112
Basis......Page 123
Dimension......Page 129
Vector Spaces and Linear Systems......Page 134
Combining Subspaces*......Page 141
Topic: Fields......Page 151
Topic: Crystals......Page 153
Topic: Voting Paradoxes......Page 157
Topic: Dimensional Analysis......Page 162
Definition and Examples......Page 169
Dimension Characterizes Isomorphism......Page 179
Definition......Page 186
Rangespace and Nullspace......Page 194
Representing Linear Maps with Matrices......Page 204
Any Matrix Represents a Linear Map*......Page 214
Sums and Scalar Products......Page 221
Matrix Multiplication......Page 224
Mechanics of Matrix Multiplication......Page 231
Inverses......Page 240
Changing Representations of Vectors......Page 248
Changing Map Representations......Page 252
Orthogonal Projection Into a Line*......Page 260
Gram-Schmidt Orthogonalization*......Page 265
Projection Into a Subspace*......Page 270
Topic: Line of Best Fit......Page 279
Topic: Geometry of Linear Maps......Page 284
Topic: Markov Chains......Page 290
Topic: Orthonormal Matrices......Page 296
Determinants......Page 303
Exploration*......Page 304
Properties of Determinants......Page 309
The Permutation Expansion......Page 313
Determinants Exist*......Page 322
Determinants as Size Functions......Page 329
Laplace's Expansion*......Page 336
Topic: Cramer's Rule......Page 341
Topic: Speed of Calculating Determinants......Page 344
Topic: Projective Geometry......Page 347
Complex Vector Spaces......Page 357
Factoring and Complex Numbers; A Review*......Page 358
Complex Representations......Page 360
Definition and Examples......Page 361
Diagonalizability......Page 363
Eigenvalues and Eigenvectors......Page 367
Self-Composition*......Page 375
Strings*......Page 378
Polynomials of Maps and Matrices*......Page 389
Jordan Canonical Form*......Page 396
Topic: Computing Eigenvalues---the Method of Powers......Page 409
Topic: Stable Populations......Page 413
Topic: Linear Recurrences......Page 415
Propositions......Page 423
Quantifiers......Page 425
Techniques of Proof......Page 427
Sets, Functions, and Relations......Page 428